Calibrating Geophysical Predictions under Constrained Probabilistic Distributions

Machine learning (ML) has shown significant promise in studying complex geophysical dynamical systems, including turbulence and climate processes. Such systems often display sensitive dependence on initial conditions, reflected in positive Lyapunov exponents, where even small perturbations in short-term forecasts can lead to large deviations in long-term outcomes. Thus, meaningful inference requires not only accurate short-term predictions, but also consistency with the system's long-term attractor that is captured by the marginal distribution of state variables. Existing approaches attempt to address this challenge by incorporating spatial and temporal dependence, but these strategies become impractical when data are extremely sparse. In this work, we show that prior knowledge of marginal distributions offers valuable complementary information to short-term observations, motivating a distribution-informed learning framework. We introduce a calibration algorithm based on normalization and the Kernelized Stein Discrepancy (KSD) to enhance ML predictions. The method here employs KSD within a reproducing kernel Hilbert space to calibrate model outputs, improving their fidelity to known physical distributions. This not only sharpens pointwise predictions but also enforces consistency with non-local statistical structures rooted in physical principles. Through synthetic experiments-spanning offline climatological CO2 fluxes and online quasi-geostrophic flow simulations-we demonstrate the robustness and broad utility of the proposed framework.